Over in the Bayonets topic, I derailed the topic a little talking about gun nuts and weapon maintenance, but it did give me a pretty great idea.
Have hobbies available at the character creation menu, these would be small things you could pick with names like handyman or fixer or gun nut, that would provide you with small morale boosts for doing jobs related to the hobby. In this case, a handyman would enjoy building things, fixers would enjoy repairing cars and damaged weapons, gun nuts would enjoy the use and maintenance of firearms. Things like that, which would really go far in developing a character.
On a similar note, have you ever eaten a food or done something you really didn’t enjoy the first time around? Yet on repeated tries you found that it was actually kinda cool? Acquired tastes would be a similar system for that, where you may not like to clean your gun at first, over time you take pride in your discipline, even in the face of the apocalypse. Basically, repeated use of certain items would slowly build up morale, but not to the point for pre-cataclysm hobbies.